In a company's first year in operation, it made an annual profit of $$ \$ 204,000 $$. The profit of the company increased at a constant $$ 18 \% $$ per year each year. How much total profit would the company make over the course of its first 24 years of operation, to the nearest whole number?

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To find the total profit the company would make over the course of its first 24 years of operation, we can model the profit as a geometric series.

1. **Identify the known values:**
   - Initial profit in the first year, $$ P_0 = 204,000 $$
   - Annual growth rate, $$ r = 0.18 $$
   - Number of years, $$ n = 24 $$

2. **Calculate the profit for each year:**
   The profit for each year can be expressed as:
   $$
   P_k = P_0 \cdot (1 + r)^k
   $$
   where $$ k $$ is the year number (starting from 0 for the first year).

3. **Total profit over 24 years:**
   The total profit $$ T $$ over 24 years can be calculated using the formula for the sum of a geometric series:
   $$
   T = P_0 \cdot \frac{(1 + r)^{n} - 1}{r}
   $$

4. **Substituting the values:**
   We will substitute $$ P_0 = 204,000 $$, $$ r = 0.18 $$, and $$ n = 24 $$ into the formula.

Now, let's calculate the total profit.

First, we need to calculate $$ (1 + r)^{n} $$:
$$
(1 + 0.18)^{24}
$$

Then we can calculate the total profit $$ T $$.

Let's perform these calculations.
Calculate the value by following steps:
- step0: Calculate:
  $$\left(1+0.18\right)^{24}$$
- step1: Add the numbers:
  $$1.18^{24}$$
- step2: Convert the expressions:
  $$\left(\frac{59}{50}\right)^{24}$$
- step3: Use the properties of exponents:
  $$\frac{59^{24}}{50^{24}}$$
Calculate or simplify the expression $$ 204000 * ((1 + 0.18)^(24) - 1) / 0.18 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{204000\left(\left(1+0.18\right)^{24}-1\right)}{0.18}$$
- step1: Add the numbers:
  $$\frac{204000\left(1.18^{24}-1\right)}{0.18}$$
- step2: Convert the expressions:
  $$\frac{204000\left(\left(\frac{59}{50}\right)^{24}-1\right)}{0.18}$$
- step3: Subtract the numbers:
  $$\frac{204000\times \frac{59^{24}-50^{24}}{50^{24}}}{0.18}$$
- step4: Multiply the numbers:
  $$\frac{\frac{51\times 59^{24}-51\times 50^{24}}{2^{19}\times 5^{45}}}{0.18}$$
- step5: Convert the expressions:
  $$\frac{\frac{51\times 59^{24}-51\times 50^{24}}{2^{19}\times 5^{45}}}{\frac{9}{50}}$$
- step6: Multiply by the reciprocal:
  $$\frac{51\times 59^{24}-51\times 50^{24}}{2^{19}\times 5^{45}}\times \frac{50}{9}$$
- step7: Rewrite the expression:
  $$\frac{3\left(17\times 59^{24}-17\times 50^{24}\right)}{2^{19}\times 5^{45}}\times \frac{50}{9}$$
- step8: Reduce the numbers:
  $$\frac{17\times 59^{24}-17\times 50^{24}}{2^{19}\times 5^{45}}\times \frac{50}{3}$$
- step9: Rewrite the expression:
  $$\frac{17\times 59^{24}-17\times 50^{24}}{2^{19}\times 5^{45}}\times \frac{2\times 25}{3}$$
- step10: Reduce the numbers:
  $$\frac{17\times 59^{24}-17\times 50^{24}}{2^{18}\times 5^{45}}\times \frac{25}{3}$$
- step11: Rewrite the expression:
  $$\frac{17\times 59^{24}-17\times 50^{24}}{2^{18}\times 5^{45}}\times \frac{5^{2}}{3}$$
- step12: Reduce the numbers:
  $$\frac{17\times 59^{24}-17\times 50^{24}}{2^{18}\times 5^{43}}\times \frac{1}{3}$$
- step13: Multiply the fractions:
  $$\frac{17\times 59^{24}-17\times 50^{24}}{2^{18}\times 5^{43}\times 3}$$
- step14: Multiply:
  $$\frac{17\times 59^{24}-17\times 50^{24}}{3\times 2^{18}\times 5^{43}}$$
1. **Calculate $$ (1 + r)^{n} $$:**
   $$
   (1 + 0.18)^{24} \approx 53.109006
   $$

2. **Calculate the total profit $$ T $$:**
   $$
   T = 204000 \cdot \frac{(1 + 0.18)^{24} - 1}{0.18}
   $$
   Substituting the value we found:
   $$
   T \approx 204000 \cdot \frac{53.109006 - 1}{0.18}
   $$

Now, let's calculate this final expression.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{204000\left(53.109006-1\right)}{0.18}$$
- step1: Subtract the numbers:
  $$\frac{204000\times 52.109006}{0.18}$$
- step2: Multiply the numbers:
  $$\frac{1.063024\times 10^{7}}{0.18}$$
- step3: Divide the terms:
  $$5.905687\times 10^{7}$$
The total profit $$ T $$ over the course of the first 24 years of operation is approximately:
$$
T \approx 5.905687 \times 10^{7}
$$

To express this in whole numbers, we round it:
$$
T \approx 59056870
$$

Thus, the total profit the company would make over the course of its first 24 years of operation is approximately **$59,056,870**.
The company would make a total profit of approximately \$59,056,870 over the first 24 years of operation.

In a company's first year in operation, it made an annual profit of \( \$ 204,000 \). The profit of the company increased at a constant \( 18 \% \) per year each year. How much total profit would the company make over the course of its first 24 years of operation, to the nearest whole number?

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